Introduction

Science is the belief in the ignorance of experts.

Richard Feynman, What is Science? Fifteenth annual meeting of the National Science Teachers Association in 1966

The Earth rotation variations The rotation of the Earth not only rules social and biological life, but it is also at the crossroads of many scientific disciplines encompassing biology, geophysics and astronomy. Indeed, the rotation of the Earth determines biological cycles at quasi- daily periods (circadian cycles), our perception of the sky, duration of the ocean tides, and many geophysical processes like cyclone formation, oceanic currents, and the magnetic field. Owing to its tiny variability, almost imperceptible to our senses, con- cerning both the angular rate and the direction of the rotation axis, rotation of the Earth arouses great interest. First, for practical reasons: variations of the rotation of the Earth day after day modify astrometric pointing at a given sidereal instant and then influence measurements done by space geodetic techniques; processing these measurements, for instance for deriving the orbits of the implied satellite or for doing ground positioning, needs accurate estimates of these variations. More fundamentally, the Earth’s rotation changes reflect global geophysical properties and processes within the Earth. Thus, by analyzing the observed fluctuations, we can better come to know our planet. The progressive discovery of these fluctuations has a long history. In terms of observation techniques, three epochs can be distin- guished. The first, from Antiquity to the early classical science in the seventeenth century, is that of astrometric pointing to the naked eye, using instruments made of wood or metal (quarter circle for example). In the seventeenth century begins the telescopic age, benefiting a double technological breakthrough: angular measure- ments are not only much more precise, but they are dated more accurately thanks to Huygens invention of pendulum clocks, regulated by a stable pendulum period. This second era ended around 1960 with the advent of space and atomic clocks tech- nology: the astrometric pointing were abandoned in favor of ultra-precise measure- ments of flight time or frequencies of electromagnetic signals propagating over Earth scale distances.1 These technological advances, combined with the development of Newtonian mechanics first revealed the astronomical nature of the fluctuations of the Earth rotation, and at the end of nineteenth century interference of geophysical causes.

1 Or difference of flight time for the VLBI technique, yielding the time delay of a given radio wavefront at two remote antennas. https://doi.org/10.1515/9783110298093-203 XX | Introduction

The Earth’s rotation: an astronomical theme until the nineteenth century By the end of the nineteenth century rotation of the Earth was still an astronomical discipline. As the precision of the angular measurements verged on 0.1 (about 1000 times the resolution power of human eye), the unique variation was the耠耠 precession– nutation of the rotation axis with respect to the starry sky. The precession—this gradual shift in the orientation of the Earth axis of rotation, which, similar to a precessing top, traces out a cone in a cycle of approximately 25800 years with a declination of 23°26 with respect to the ecliptic pole axis—had been discovered from the Antiquity (around耠 200 B.-C. by the Greek Hipparchus); the nutation—a composition of periodic oscillations with amplitude smaller than 20 and superimposed to the precession— was unveiled thanks to telescopic astrometry;耠耠 Bradley discovered its main component of 18.6 years in 1748. In the light of the new mechanics of Newton, precession and nutation result from the gravitational lunisolar forces on the Earth’s equatorial bulge. Even if the amplitude of each term is proportional to the Earth’s dynamical ellipticity, its cause is nonetheless astronomical.

Geophysical breakthrough By the beginning of the nineteenth century Laplace investigated the influence of vari- ous terrestrial causes on the Earth’s angular velocity. However, the order of magnitude of the estimated effects was out of reach of the observation of his2 epoch. The geophys- ical breakthrough came where it was least expected. In 1750 Euler enlarged Newtonian mechanics to extended solid bodies, and from its new theorems could prove that the rotation pole freely moves with respect to the Earth’s surface with a period equal to the sidereal period divided by the Earth dynamical ellipticity [84].3 During 150 years, this polar motion was actively looked for in astronomical latitudes (angle between the true equator, perpendicular to rotation axis, and the local vertical). Finally, in 1891, Chandler [34, 35] related an oscillation of about 0.2 with a period of 430 days, not of 305 days, as expected. One year later, Newcomb耠耠 [162] explained how Earth’s non rigidity can lengthen the Euler period of 130 days. This allowed him to state that the solid Earth had an elasticity comparable to the steel one. From that moment Earth’s rotation became a means for inferring geophysical properties. In the same time an an- nual oscillation was discovered, with the half amplitude of the Chandler wobble [36]. So, after Lord Kelvin [119], Newcomb [163] and other scientists naturally assumed the role of seasonal air and water circulation in polar motion.

2 “J’ai discuté dans le cinquième livre de la Mécanique céleste l’influence des causes intérieures telles que les volcans, les tremblements de terre, les vents, les courants de la mer, etc., sur la durée de rota- tion de la terre; et j’ai fait voir au moyen du principe des aires que cette influence est insensible (…)” P.S. de Laplace, Exposition du système du monde, Sixth ed. (1827) p. 344. 3 Euler determined a flattening of about 1 230 from the available measurement of the epoch [83], so he found a period of about 230 sidereal days against 305 for the contemporaneous value of the so-called Euler period. / Introduction | XXI

In the vein of a pioneering study by Hopkins in 1839 [76], Hough, Sloudsky and Henri Poincaré theorized the effect of a fluid core, still hypothetical, on the Earthro- tation at the dawn of the twentieth century. Assuming incompressible, homogeneous and non-viscous fluid contained in an ellipsoidal rigid cavity, Hough [116] and Sloud- sky [202] independently predicted a second free oscillation of the rotation pole, quasi- diurnal and retrograde in the terrestrial frame, resulting in a celestial nutation, of which the period is inversely proportional to the flattening of the fluid core. In 1910, Poincaré [172] showed how the terms of the nutation resonate at this period and there- fore differ from those of a rigid Earth.

Discovery of variations in the rotation angular rate In the early twentieth century, it was proved that for the Earth the axis of rotation wobbles, but the diurnal rotation constituted a flawless clock until the 1920s: the suc- cession of seconds of the mean solar day (representing conventionally 1.00273 times the stellar rotation period assumed to be constant) realized the Universal Time (UT) with the convention that the mean sun passes the meridian of Greenwich at 12 hour UT. But, following the work of Newcomb and Spencer Jones (1926) [208], there was rec- ognized in the Length Of Day (LOD) a secular growth of about +1.6 ms per century and decadal fluctuations of a few milliseconds, which are interpreted by the dissipation ac- companying tidal deformations and core–mantle coupling, respectively. Then, in the 1930s, comparing UT with the time of quartz clocks which had just been developed, Stoyko found seasonal variations of about 20 ms, namely 0.5 ms in LOD, and it was immediately thought of in terms of the impact of seasonal atmospheric circulation.

Advent of the Space Age and consequences In the period following the discovery of polar motion, instrumentation did not make substantial progress, and the quality of optical surveys was still affected by atmo- spheric turbulence. But regular observations of the latitude were then carried out over several observatories around the world to determine the terrestrial oscillations of the pole. And from 1900 to 1960, the temporal resolution going from months to weeks, the accuracy from grew from 0.03 to 0.01 . In the 1960s began the Space Age. Satel- lites, providing measurements of耠耠 the whole耠耠 Earth’s surface, enabled the determina- tion of overall physical properties of the solid Earth and its fluid envelope, in addi- tion to balloon or local surface measurements. Processing an increasing quantity of information was then made possible thanks to growing computer capabilities. This resulted in considerable progress in the measurement and modeling of meteorolog- ical, oceanographic and hydrological hazards. Analysis of satellite orbital perturba- tions also revolutionized our knowledge of the Earth’s gravity field on a large scale. From this epoch, “planetary” technology, such as Very Long Baseline Interferometry (VLBI), was developed to determine the shape of the Earth, its gravity field and its motions, and gradually replaced optical astrometry for ground positioning as well as XXII | Introduction

for geodynamic studies. Space geodesy was born. In the 1980s, it addressed global deformations like tectonic drift. In 1985, it permitted to determine the Earth’s rotation irregularities 10 times more precisely than the optical astrometry does; this technique, downgraded, was abandoned in geodynamic studies and geodetic applications. This concomitant progress in the knowledge of Earth’s surface movements and mass re- distributions operating at its surface, or within it, confirmed the hypotheses made at the end of the nineteenth century. First, polar motion is indeed influenced by the at- mosphere, a little less by the oceans and inland waters (ice, snow, soil moisture and vegetation included). Second, from the 1980s the VLBI observations clearly have un- veiled the resonance of the lunisolar terms of the nutation at a 430 day period, in 1986 a non-tidal nutation was discovered at this period and immediately attributed to the corresponding free mode.

Decoupling the rotational speed from the polar motion Over periods smaller than one century, changes in the rotation of the Earth are small enough to be treated as perturbations. The law of angular momentum written in the terrestrial reference system then provides a set of first order linear differential equa- tions, the Liouville linear equations decoupling angular rotation velocity (axial per- turbation) from rotation axis direction (equatorial perturbation involving polar mo- tion and nutation). In these equations the various geophysical and astronomical ef- fects can be calculated separately. Summing them, we obtain the total effect which can be compared with the observations. Often this summation is not necessary be- cause the studied effect has such a particular signature that it is easily recognizable in the observed variation and cannot be confused with the imprint of another phe- nomenon.

Nature of axial and equatorial excitations For periods of less than 10 years, changes in LOD (1 ms amplitude) primarily result from tidal deformations and atmospheric winds. Beyond 10 years, the variations are larger, and they are commonly attributed to the interaction between the core and the mantle. However, the modeling of this internal mechanism is widely speculative and cannot be verified by direct observations. In contrast, polar motion originates from the surface fluid layer or the lithosphere, that is to say, superficially with regard tothe diameter of the Earth.4 They are observable, at least estimable, in contrast to internal processes for which there is only indirect observations like magnetic field traducing mass transport within the core as far as dynamo model is sound. So, looking at their causes, polar motion and length of day variation differ substantially.

4 With the exception of the Markowitz term (20–30 years), which may result from the core-mantle interaction. Introduction | XXIII

Purpose of this book These considerations lead us to treat polar motion, and in a broader sense nutation, irrespective of the LOD variation. Polar motion remains better measured than it is understood. The compilation and analysis of historical observations can trace it since the middle of the nineteenth century. Over longer periods, the secular drift at the rate of about 0.4 /century, is larger than the Chandler and seasonal wobble with a total amplitude not耠耠 exceeding 0.4 . This secular trend not only comes from mass redistribution but it can eventually耠耠 result from the continental drift. If all stations, owing to their lithospheric drift, rotate on the mantle, the pole will shift accordingly. So in the drift will overlap both tectonic effect and changes of the moment of iner- tia, such as those caused by the mountain uplift or ice melting. Some observations, such as fossilized tropical plants in now temperate areas, demonstrate a shift in the pole of rotation relative to the continents, but it is difficult to distinguish the tectonic shift from the effects of mass transports. But we only evoke this distant pastinthe last chapter, for we shall focus on the contemporaneous or astrometric polar motion, started about 150 years ago. This time window allows us to embrace any oscillation from a few hours to hundred years. The purpose of this book is to model them in the light of global geophysical processes, as we modeled them today. Sometimes the theory is so complex that we cannot see its relevance to interpret the observation; conversely the usual theoretical formalism cannot be fully consis- tent with the present accuracy of Earth rotation determination. So, we endeavor to tie the theoretical modeling to the uncertainty that spoils the geophysical excitation. By stressing this “metrological” aspect, we hope to complement the books of Sidorenkov [200, 201], throwing meteorological light on the instabilities of the Earth rotation. Be- sides seasonal, daily and half-daily variations, polar motion has a strong stochastic part, making it hardly predictable in contrast to the lunisolar precession–nutation. In this view, this work is a logical extension of the recent book of Dehant and Mathews [60], mostly devoted to the regular wobble of the rotation axis,5 namely the luniso- lar precession–nutation. The description of the Earth response to an external poten- tial through the Love numbers formalism is a cornerstone of the polar motion theory. At contemporaneous time scale, this question is treated in the book by Dehant and Mathews, but more comprehensively in the book by Symlie (2013) Earth dynamics: deformations and oscillations of the Rotating Earth [204], which also brings precious insight for investigating the possible seismic effect on rotation of the Earth. Whereas it is mostly focused on the sub-secular polar motion, this synthesis finally touches on the polar motion at secular and geological time scales. Then, the quasi-elastic solid Earth has to be given up for a visco-elastic Earth. For completing our short review of

5 In a general sense, encompassing the observed rotation axis or celestial intermediate axis, the in- stantaneous rotation axis, the figure axis or the angular momentum axis. All these notions will be pre- cised later. XXIV | Introduction

Figure 1: Study of polar motion. this problem, we recommend the more comprehensive book of Sabadini, Vermeersen and Cambiotti [178]. In tracking and modelling polar motion, the space-time reference systems are cornerstones, of which the reader will find an updated and comprehensive presentation in the book of Soffel and Langhans [205]. The other references go back to the 1980s. Munk and MacDonald (1960) [153], Lambeck (1980) [126], and Moritz and Mueller (1987) [152] contain much valuable material, but these references have not not benefited from the advances of space geodesy and global circulation models of the hydro-atmosphere.

Contents While this book presents the polar motion studies with its most recent developments, it is addressed not only to specialists but also to a wide public not having prerequi- site knowledge in geophysics or astrogeodesy. The reader will discover or deepen the intricacies of the polar motion, studied in light of a dynamical system: i) the input, namely the geophysical excitation, ii) the transfer operated by the Earth system and iii) the output, namely the polar motion (see Figure 1). First we give an overview of the system output, that is to say, changes in the ro- tation of the Earth. After exposing the basic aspects (space time reference systems, Chapter 1), we give a synthetic outlook of the Earth rotation variations (Chapter 2), and we describe the way they are parametrized and determined by processing astro- geodetic measurements (Chapter 3). In the second part we formulate how the Earth system transmits the mechanical excitation to polar motion. After giving the dynam- ical foundations (Chapter 4), we consider the effect of the rotational deformation of the mantle (Chapter 5) and of the ocean surface (Chapter 6) accompanying the polar Introduction | XXV

motion, and model the influence of the fluid core (Chapter 7). In the light of the current observation accuracy, a biaxial Earth model and other theoretical approximations are insufficient. Accordingly triaxiality and the asymmetric effect of the oceans arecon- sidered and lead us to introduce the formalism of generalized Liouville equation in the equatorial plane (Appendix A). Finally, the third part is devoted to the study of geophysical excitation (mainly produced by the mass redistribution) and its effect on polar motion within the theoretical framework of the second part and for time scales ranging from 12 hours to a few decades. Atmospheric, ocean, hydro-continental exci- tations are described (Chapter 8), they are compared globally to polar motion (Chap- ter 9), and then in more detail for characteristic frequency bands of the contempora- neous polar motion: rapid, annual, inter-annual, and decadal (Chapter 10), Chandler band (Chapter 11), diurnal and sub-diurnal (Chapter 12), and retrograde diurnal band (Chapter 13). This allows us to clarify some outstanding issues such as the origin of the Chandler term (Chapter 11), the impact of inland fresh waters (Chapter 9), diur- nal and subdiurnal variations (Chapter 12), and their effect on nutation (Chapter 13). Finally, extending our considerations even further, we deal with transient or jerk-like excitation by modeling the seismic contributions (Chapter 14), and treat the polar drift produced by the largest known mass redistribution, occurring at prehistoric or geolog- ical time scales (Chapter 15), and then determined by a visco-elastic mantle.